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Arzelà’s dominated convergence theorem for the Riemann integral. (English) Zbl 0863.26010

The authors present first of all another elementary proof of Arzelà’s dominated convergence theorem by simplifying the reviewer’s proof of the main lemma that a decreasing sequence of bounded real functions defined on a bounded and closed interval that decreases everywhere to zero has the property that their lower Darboux integrals tend to zero. The authors also present a completely different proof of the main lemma that is based on Cousin’s lemma concerning the existence of special partitions of an interval. The authors point out that this result is equivalent to the completeness of \(\mathbb{R}\), but so is Dini’s theorem that is used in the other version of the proof of the main lemma. In the final section, the authors show a restricted version of the not-too-well known fact that the lower Darboux integral has the Fatou property. This is a consequence of the main lemma, the superadditivity property of the lower Darboux integral and the fact that length is a countably additive measuring function (which is equivalent to the completeness property of \(\mathbb{R}\)).

MSC:

26A42 Integrals of Riemann, Stieltjes and Lebesgue type
28A25 Integration with respect to measures and other set functions
26A39 Denjoy and Perron integrals, other special integrals
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